RFC on Sparse matrices in R

Roger Koenker and Pin Ng have provided a sparse
matrix implementation for R in the SparseM
package, which is based on Fortran code in sparskit
and a modified version of the sparse Cholesky factorization
written by Esmond Ng and Barry Peyton. The modified version is
distributed as part of PCx by Czyzyk, Mehrotra, Wagner, and
Wright and is copywrite by the University of Chicago.

Recently I become very interested in certain sparse matrix
calculations myself and have looked at some of the available
Open Source software for the sparse Cholesky decomposition.
While I certainly appreciate the work that Roger and Pin have
done I will propose a slightly different implementation.

Representations of sparse matrices

Conceptually, the simplest representation of a sparse matrix is
as a triplet of an integer vector i giving the row
numbers, an integer vector j giving the column numbers,
and a numeric vector x giving the non-zero values in the
matrix. An S4 class definition might be

The triplet representation is row-oriented if elements in
the same row were adjacent and column-oriented if elements in the
same column were adjacent. The compressed sparse row (csr)
(or compressed sparse column - csc) representation is
similar
to row-oriented triplet (column-oriented triplet) except that
i (j) just stores the index of the first element
in the row (column). (There are a couple of other details but
that is the gist of it.) These compressed representations
remove the redundant row (column) indices and provide faster
access to a given location in the matrix because you only need
to check one row (column).

The preferred representation of sparse matrices in the SparseM
package is csr. Matlab
uses csc. We hope that Octave will also use this
representation. There are certain advantages to csc in systems
like R and Matlab where dense matrices are stored in
column-major order. For example, Sivan Toledo's TAUCS library and
Tim Davis's UMFPACK
library are both based on csc and can both use level-3 BLAS in
certain sparse matrix computations.

I feel that compatibility with Matlab (and, we hope, Octave), the
ability to use level-3 BLAS, and the availability of the
csc-based TAUCS,
UMFPACK, and AMD
libraries favors csc as the preferred sparse matrix representation.

Applications of sparse matrices

I imagine that the main applications of sparse matrices in R will
be for parameter estimation in very large linear models and for
large sparse contingency tables.

As Roger and Pin have pointed out, the key to estimating
parameters in large linear models quickly and with minimal
storage requirements will be in providing a way for
model.matrix to generate a sparse model matrix
X or a sparse symmetric representation of
X'X and X'y.

Assuming that we have a sparse representation of the model
matrix as mm and a sparse or dense representation
of the response as y, the coefficients can be
estimated as

solve(crossprod(mm), crossprod(mm, y))

I think that the multifrontal sparse Cholesky in the TAUCS
library is one of the best currently available ways to do this
and have implemented a solution based on that in the
taucs package for R. I use the approximate minimal
degree ordering determined by Tim Davis's AMD library to reduce
fill-in.

For statistical analysis of a linear model we probably also want
at least the standard errors of the coefficient estimates which
means we want an inverse of the Cholesky factor. TAUCS has an
inverse factorization routine taucs_ccs_factor_xxt
that can provide a sparse representation of the inverse. I
think that we want to use that for a linear model analysis. We
can use the multifrontal solver if we only want coefficients.

When working with linear models there will be a tradeoff between
the speed boost available by reordering rows and columns and the
statistical information available in the original ordering of
the rows and columns. For example, the simplest way to
determine the sequential sums of squares of the terms in the
model is to maintain the column ordering in X but
that could result in dramatic amounts of fill-in for the sparse
Cholesky and especially for the inverse factorization. I think
it is best to compromise and obtain the inverse factorization of
the reordered matrix. This can provide standard errors and
correlations of coefficients but not the sequential sums of
squares. (At least I don't know how to get them from the
reordered matrix.)

Sparse contingency tables can be easily constructed and
manipulated. I understand that Kurt Hornik would
like to use them but I'm not sure exactly what operations he needs.

I have a hybrid application involving large linear mixed models
with partially
crossed grouping factors. For these I need to manipulate
both sparse contingency tables and some associated sparse positive
definite matrices.

Utilities for sparse matrices

TAUCS has a convenient C struct for the csc representation of a
matrix. I have written functions to transfer from an S4 object
to the TAUCS struct and back. TAUCS also has routines for
multiplication by dense matrices and for symmetric permutation
of rows and columns (needed in the Cholesky factorization
routines).

UMFPACK is a set of routines for solving unsymmetric sparse
linear systems with the Unsymmetric MultiFrontal method. It has
a couple of very convenient routines for switching between csc
and a triplet representation. The triplet to csc converter is
quite general in that it allows redundant triplet
representations (more than one entry for the same position -
multiple entries have their values summed) and arbitrary
ordering. This allows convenient creation of sparse contingency
tables (build up the triplet representation then compress it).
As described in the UMFPACK documentation, it also allows simple
ways to write operations like transposition of matrices (convert
csc to triplet, interchange i and j, convert back
to csc).

As a side note, it appears that the UMFPACK/AMD form of the csc
representation is more strict than the TAUCS representation. If
one applies taucs_ccs_permute_symmetrically to a
csc matrix (in TAUCS these are called ccs) the result does not
have the rows in increasing order within each column. AMD
doesn't like this and I find it confusing when trying to examine
the matrix. Again the csc to triplet to csc conversion can be
used to remove this problem.

Licenses

TAUCS is available under the LGPL. UMFPACK is official
GNU software covered by what is described as a 2-clause BSD
license. I'm not sure exactly what that means. AMD is covered
by a similar license and I think may be considered part of
UMFPACK in the sense of also being official GNU software. The
tarball for UMFPACK contains AMD as you need AMD to be able to
use UMFPACK.

Proposed plan

Build a library using S4 classes and methods for the csc
representation and based on the TAUCS code for the sparse
Cholesky plus AMD ordering. I would include at least the csc
to triplet utility routines from UMFPACK. Depending upon
whether there was substantial demand for unsymmetric (LU)
sparse matrix factorization and solution of systems, we could
bring in the whole of UMFPACK or look at the unsymmetric
solvers in TAUCS.

Modify model.matrix to produce a sparse
representation of X or of X'X and
X'y. It would be convenient to get a sparse
representation of the model matrix but the big payoff would be
in getting the crossproduct matrix. However, it is difficult
to decide where the non-zero elements in the crossproduct
matrix are when you are sequentially examining the rows of
X. It may be possible to perform the operation
on chunks of rows and use the csc to triplet to csc
transformation if new non-zero elements are found while
processing a chunk.